{
  "abstract": "We study bond percolation on the one-dimensional lattice in which two sites at distance $r$ are connected with probability $p_r=C/r^{1+σ}$, where $0 < σ< 1$. An infinite cluster exists if $C$ exceeds a critical value $C_c(σ)$, which is not known exactly: the available results are the lower bound of Schulman, $C_c\\ge1/[2ζ(1+σ)]$, and numerical estimates. Here we prove explicit upper bounds. We apply a second-moment argument to a family of random monotone paths whose steps have a heavy-tailed (Sibuya) distribution. Two such paths meet only a finite number of times, and their overlap is computed exactly by renewal theory. The result is a closed-form bound, which gives $C_c\\leσ+O(σ^2)$ as $σ\\to0$ and is below $1$ for $σ< 0.667$. Since the paths are monotone, the bound holds also for oriented percolation, and it determines the oriented threshold asymptotically: $C_c^{\\rightarrow}ζ(1+σ)\\to1$ as $σ\\to0$. A coarse-grained version of the argument, with blocks of sites in place of sites, gives an explicit bound $C_c < 1$ for every $σ< 1$. We compare the bounds with the known numerical values of $C_c$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11872",
  "author_total": 1,
  "authors": [
    {
      "name": "Guy Amit"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f46z31t9zacy0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11872",
  "fulltext_url": null,
  "id": "f46z31t9zacy0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11872",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7221111732",
  "pdf_url": "https://arxiv.org/pdf/2610.11872",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f46z31t9zacy0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:54Z",
      "native_id": "2610.11872",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11872"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Explicit upper bounds on the threshold of one-dimensional long-range percolation",
  "title_text": "Explicit upper bounds on the threshold of one-dimensional long-range percolation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/f46z31t9zacy0",
  "venue": "arXiv",
  "versions": []
}